The Black Cats will be bottom of the table at Christmas, regardless of the outcome of Saturday's home clash with Norwich.
Only one team in that position in the history of the Premier League has gone on to avoid the drop - West Brom in 2005 - but Poyet is refusing to give up the fight.
While nine players, including the likes of Jack Colback, Keiren Westwood, Phil Bardsley, Craig Gardner and Seb Larsson, will be out of contract in the summer and three more are on loan, he expects them all to play to the 'last minute' in the battle to avoid the drop.
Poyet said: "We are in a situation where we are fighting for our lives. We have nine players finishing contracts, three that are on loan and 12 that have just arrived and are all foreigners.
"We are managing this situation, although it is not easy. It's not easy for the player, firstly. They want to know their futures as soon as possible. Can we commit now to most of those players? No, we cannot. We don't know where we are going to be.
"There are some clubs where they will put a player under pressure to sign 18 months from the end of his contract or they won't play anymore.
"I'm a manager who believes if you sign for three years, you play for three years and we use you to the last minute of the last game. Then if you're free, you're free.
"I trust professionals will go to the last minute with this season. I will go with the team that I think it is the best. If that includes players who will be out of contract, that's fine.
"Regardless of whether we go down or stay up, I trust the players every day. I'm convinced they will all be committed to the cause regardless of their situation. You can see if a player is still committed or not - and if I have any doubts he won't play. As it stands, I have had no problems at all."
Poyet still believes his side can stay up, citing an improvement in performances which the Black Cats must now turn into tangible points.
He added: "Performance-wise, we have been improving, but in terms of points, no.
"I'm very honest. It's easier to win when you perform well and if we maintain consistency in our performance, we have a great chance of staying up.
"There are no excuses (against Norwich) because confidence is high, the players are fit and we are playing at home."